Two small plastic balls hang from threads of negligible mass. Each ball has a mass of 0.14g and a charge of magnitude q. the balls are attracted to each other. When in equilibrium, the balls are separated by a distance of 2.05cm and the threads attached to the balls make an angle of 20 degrees with the vertical.
Find:
a. The magnitude of the electric force acting on each ball.
b. The tension in each of the threads.
c. The magnitude of the charge on the balls.
r=0.0205 m
α=20⁰,
m = 0.00014 kg=1.4•10⁻ ⁴ kg•
F(el)=kq₁•q₂/r²=kq²/r²,
T•sin α =F(el)= kq²/r²,
T•cosα=mg .
T•sin α/ T•cosα= kq²/r²mg,
tanα = kq²/r²mg,
q=sqrt{ r²•m•g•tan α/k}= ...
T= mg/cos α = ….
1.0736 x10^-7
To solve this problem, we can use the concepts of electrostatics and forces in equilibrium. Let's break it down step by step:
Step 1: Calculate the weight of each ball.
The weight of each ball can be given by the formula:
Weight = mass × acceleration due to gravity (g)
Given that each ball has a mass of 0.14g, and the acceleration due to gravity is approximately 9.8 m/s^2, we can calculate the weight for each ball:
Weight = 0.14g × 9.8 m/s^2
Step 2: Calculate the vertical component of the weight.
The vertical component of the weight can be calculated using the formula:
Vertical Component = Weight × sin(angle)
Given that the threads make an angle of 20 degrees with the vertical, we can calculate the vertical component of the weight for each ball:
Vertical Component = Weight × sin(20°)
Step 3: Calculate the tension in each thread.
In order for the balls to be in equilibrium, the tensions in the threads must balance the vertical components of the weight. Therefore, the tension in each thread is equal to the vertical component of the weight. So:
Tension = Vertical Component
Step 4: Calculate the electric force acting on each ball.
The electric force between the two charged balls can be calculated using Coulomb's Law:
Electric Force = (k × q^2) / r^2
where k is the electrostatic constant, q is the charge on each ball, and r is the distance between the balls.
Given that the balls are in equilibrium, the electric force on each ball should balance the weight. So:
Electric Force = Weight = mass × acceleration due to gravity
Step 5: Calculate the magnitude of the charge on the balls.
Since the electric force on each ball is the same and equal to the weight, we can set up the equation:
(mass × acceleration due to gravity) = (k × q^2) / r^2
Rearranging the equation gives us:
q^2 = (mass × acceleration due to gravity × r^2) / k
Taking the square root on both sides, we get:
q = √((mass × acceleration due to gravity × r^2) / k)
Now, let's plug in the given values and solve these equations to find the desired quantities:
Step 1: Calculate the weight of each ball:
Weight = 0.14g × 9.8 m/s^2
Step 2: Calculate the vertical component of the weight:
Vertical Component = Weight × sin(20°)
Step 3: Calculate the tension in each thread:
Tension = Vertical Component
Step 4: Calculate the electric force acting on each ball:
Electric Force = Weight = mass × acceleration due to gravity
Step 5: Calculate the magnitude of the charge on the balls:
q = √((mass × acceleration due to gravity × r^2) / k)
Finally, substitute the given values to get the numerical answers.
To solve for the magnitude of the electric force acting on each ball, you need to use Coulomb's Law. Coulomb's Law states that the magnitude of the electric force between two charged objects is proportional to the product of their charges and inversely proportional to the square of the distance between them.
a. To find the magnitude of the electric force acting on each ball, use the formula:
F = k * (q₁ * q₂) / r²
Where:
F is the electric force,
k is the electrostatic constant (k = 9 × 10^9 Nm²/C²),
q₁ and q₂ are the charges on the balls, and
r is the distance between the balls.
Plugging in the known values:
q₁ = q₂ (since the balls have the same charge),
r = 2.05 cm = 0.0205 m,
The formula becomes:
F = (9 × 10^9 Nm²/C²) * (q²) / (0.0205 m)²
b. To find the tension in each of the threads, you need to analyze the forces acting on the balls when they are in equilibrium. In this case, the forces acting on each ball are the weight and the tension in the thread.
When in equilibrium, the vertical components of the forces acting on the balls must balance. The tension in each thread can be written as:
Tension = Weight × cos(θ)
Where:
Weight is the gravitational force acting on each ball (Weight = mass × g),
mass is the mass of each ball (mass = 0.14g = 0.00014 kg),
g is the acceleration due to gravity (g = 9.8 m/s²), and
θ is the angle the threads make with the vertical (θ = 20 degrees).
c. To find the magnitude of the charge on the balls, you need to relate it to the electric force and the tension. Equating the electric force and the tension, you get:
F = Tension
By substituting the respective formulas for F and Tension, you can solve for q.
Now you have the general understanding of how to approach this problem. You can proceed by plugging in the known values and solving for each unknown.